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The theory of P\'olya ensembles of positive definite random matrices providesstructural formulas for the corresponding biorthogonal pair, and correlationkernel, which are well suited to computing the hard edge large $N$ asymptotics.Such an analysis is carried out for products of Laguerre ensembles, theLaguerre Muttalib-Borodin ensemble, and products of Laguerre ensembles andtheir inverses. The latter includes as a special case the Jacobi unitaryensemble. In each case the hard edge scaled kernel permits an expansion inpowers of $1/N$, with the leading term given in a structured form involving thehard edge scaling of the biorthogonal pair. The Laguerre and Jacobi ensembleshave the special feature that their hard edge scaled kernel -- the Besselkernel -- is symmetric and this leads to there being a choice of hard edgescaling variables for which the rate of convergence of the correlationfunctions is $O(1/N^2)$.

Rate of convergence at the hard edge for various P\'olya ensembles of positive definite matrices